Modal Expansion

A special case of transfer-function modeling is known as modal synthesis [8,9,145,385]. It can also be seen as the basis for formant synthesis and the source-filter synthesis model (resonators excited by various means). In this technique, the physical system is characterized in terms of its resonant modes. It finds extensive application in industry for determining parametric frequency responses from measured vibration data. Each mode is typically described in terms of its resonant frequency, bandwidth (or damping), and gain (and perhaps phase).

Since the modal parameters specify the spectral formants of the system, modal synthesis can be regarded as including formant synthesis so often applied to the synthesis of voice [222,264,395,39,84,504,503,321,372,301]. Since the importance of spectral formants in sound synthesis has more to do with the way we hear than with the physical parameters of a system, formant synthesis is best viewed as spectral modeling synthesis techniques as opposed to a true physical modeling technique [445]. An exception to this rule may occur when the physical system truly consists of a parallel bank of second-order resonators, such as an array of tuning forks or Helmholtz resonators. In that case, the mode parameters correspond to to physically independent objects. However, this is rare in practice.

Since only the modes in the range of human hearing need be retained, and since ``uncontrollable'' or ``unobservable'' modes can be left out, the modal representation is generally more efficient than an explicit mass-spring-dashpot digitization such as obtained using wave digital filters. On the other hand, since the modal representation normally sacrifices a physical description, it may be difficult or impossible to add nonlinearities that behave naturally.

Given any order